Automorphic Factor

In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article ' factor of automorphy'.

Definition

An ''automorphic factor of weight k'' is a function
$\nu : \Gamma \times \mathbb \to \Complex$
satisfying the four properties given below. Here, the notation $\mathbb$ and $\Complex$ refer to the upper half-plane and the complex plane, respectively. The notation $\Gamma$ is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element $\gamma \in \Gamma$ is a 2×2 matrix
$\gamma = \begina&b \\c & d\end$
with ''a'', ''b'', ''c'', ''d'' real numbers, satisfying ''ad''−''bc''=1.

An automorphic factor must satisfy:
# For a fixed $\gamma\in\Gamma$, the function $\nu(\gamma,z)$ is a holomorphic function of $z\in\mathbb$.
# For all $z\in\mathbb$ and $\gamma\in\Gamma$, one has $\vert\nu(\gamma,z)\vert = \vert cz + d\vert^k$ for a fixed real number ''k''.
# For all $z\in\mathbb$ and $\gamma,\delta \in \Gamma$, one has $\nu(\gamma\delta, z) = \nu(\gamma,\delta z)\nu(\delta,z)$ Here, $\delta z$ is the fractional linear transform of $z$ by $\delta$.
# If $-I\in\Gamma$, then for all $z\in\mathbb$ and $\gamma \in \Gamma$, one has $\nu(-\gamma,z) = \nu(\gamma,z)$ Here, ''I'' denotes the identity matrix.

Properties

Every automorphic factor may be written as

:$\nu(\gamma, z)=\upsilon(\gamma) (cz+d)^k$

with
:$\vert\upsilon(\gamma)\vert = 1$

The function $\upsilon:\Gamma\to S^1$ is called a multiplier system. Clearly,

:$\upsilon(I)=1$,
while, if $-I\in\Gamma$, then

:$\upsilon(-I)=e^$
which equals $(-1)^k$ when ''k'' is an integer.

References

* Robert Rankin, ''Modular Forms and Functions'', (1977) Cambridge University Press {{ISBN, 0-521-21212-X. ''(Chapter 3 is entirely devoted to automorphic factors for the modular group.)''

Modular forms